A man invested a certain sum in scheme A at 15% p.a. for 2 years and earned Rs 1950 as simple interest. He increased his sum by Rs. *x* and invested in another scheme B at 10% p.a. compound interest for 2 years and received Rs. 1680 as compound interest. Find the value of *x*?

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**Solution:**

1. Let the original sum invested in scheme A be *P*.

2. Given simple interest (SI) for 2 years at 15% p.a. is Rs 1950.

Using the formula for simple interest:
   $$SI = \frac{P 	imes R 	imes T}{100}$$

Substitute values:
   $$1950 = \frac{P 	imes 15 	imes 2}{100}$$

3. Solve for *P*:
   $$P = \frac{1950 	imes 100}{15 	imes 2} = \frac{1950 	imes 100}{30} = 6500$$

4. The man increases his sum by Rs *x* and invests in scheme B at 10% p.a. compound interest for 2 years.

5. Compound interest (CI) for 2 years at 10% p.a. is calculated as:

Compound amount factor:
   $$ (1 + \frac{10}{100})^2 = (1.1)^2 = 1.21 $$

Compound interest rate over 2 years = 21% (since 1.21 - 1 = 0.21 or 21%)

6. Given compound interest earned is Rs 1680, so:

$$ CI = (6500 + x) 	imes \frac{21}{100} = 1680 $$

7. Solve for *x*:

$$ (6500 + x) = \frac{1680 	imes 100}{21} = 8000 $$

$$ x = 8000 - 6500 = 1500 $$

**Answer:** Rs. 1500

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